Epidemics 1 (2009) 2–13

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Epidemics

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No coexistence for free: Neutral null models for multistrain pathogens

Marc Lipsitch a,b,⁎, Caroline Colijn a,c, Ted Cohen a,d, William P. Hanage e, Christophe Fraser f

a Department of Epidemiology, Harvard School of Public Health, Boston, MA, USAb Department of Immunology and Infectious Diseases, Harvard School of Public Health, USAc Department of Engineering Mathematics, University of Bristol, Bristol, UKd Division of Global Health Equity, Harvard Medical School, Boston, MA, USAe Department of Infectious Disease Epidemiology, Imperial College London, London, UKf Medical Research Council Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Imperial College London, London, UK

⁎ Corresponding author. Department of EpidemioloHealth, 677 Huntington Avenue, Boston, MA 02115, USA

E-mail address: mlipsitc@hsph.harvard.edu (M. Lips

1755-4365/$ – see front matter © 2008 Published by Edoi:10.1016/j.epidem.2008.07.001

a b s t r a c t

a r t i c l e i n f o

Article history:

In most pathogens, multipl Received 2 May 2008Revised 21 July 2008Accepted 30 July 2008

Keywords:Mathematical modelsStain coexistenceNeutral modelsPopulation geneticsEcologyInfectious disease epidemiology

e strains are maintained within host populations. Quantifying the mechanismsunderlying strain coexistence would aid public health planning and improve understanding of diseasedynamics. We argue that mathematical models of strain coexistence, when applied to indistinguishablestrains, should meet criteria for both ecological neutrality and population genetic neutrality. We show thatclosed clonal transmission models which can be written in an “ancestor-tracing” form that meets the formercriterion will also satisfy the latter. Neutral models can be a parsimonious starting point for studyingmechanisms of strain coexistence; implications for past and future studies are discussed.

© 2008 Published by Elsevier Inc.

A central question in infectious disease epidemiology is: whatcreates and maintains patterns of strain diversity in pathogenpopulations? This question not only holds intrinsic scientific interest,but affects a number of practical decisions. How much, and how fast,will the prevalence of antimicrobial resistance in pathogen popula-tions change in response to changes in antimicrobial use? For vaccinesthat target a subset of strains in polymorphic pathogen populations(e.g. vaccines against the malaria parasite Plasmodium falciparum,influenza virus, Streptococcus pneumoniae, human papillomavirus),how will vaccination change the patterns of pathogen diversity, andhow will these changes affect the incidence of disease? Mathematicalmodels are routinely used to try to address such questions, and anysuch model, implicitly or explicitly, includes assumptions about themechanisms that created the pathogen diversity upon which a newselective pressure may act. Confidence in the predictions of suchmodels depends on whether the mechanisms that underlie straindiversity are biologically and mechanistically plausible. The impetusfor the work described in this paper was to describe conditions thatguarantee that mechanisms in a model that promote stable co-existence are explicit (rather than hidden mechanisms that give“coexistence for free”), allowing these mechanisms to be testedagainst biological data.

gy, Harvard School of Public.itch).

lsevier Inc.

Strains may differ by neutral markers, by drug resistancedeterminants, by their expression of various antigens that triggerimmune responses, by factors affecting infectiousness or persistence,and often by a combination of these traits. Patterns of diversity may bestatic in space and time or may be dynamic, reflecting changingselective pressures and/or the effects of genetic and ecological drift(Gupta and Maiden, 2001; Grenfell et al., 2004; Rambaut et al., 2004,2008; Lipsitch and O'Hagan, 2007). Indeed, the question of straindiversity in epidemiology closely parallels the problems of standinggenetic polymorphism in population genetics (Kimura, 1985), and ofspecies diversity in community ecology (Hubbell, 2001). As in thesefields, both selective pressure and neutral drift undoubtedly play somerole, and a key problem is to distinguish these roles. A central feature ofproblems concerning diversity is the coexistence of strains. This maybe a consequence of “balancing” selective pressures maintaining eachstrain alongside one another, or it may be a feature of neutral drift.

Mathematical models are important tools both to understand themechanisms underlying strain coexistence, and to predict the impact ofpublic health interventions that exert selection on pathogen popula-tions. In ecology, the classic model of competition between two speciesis the Lotka–Volterra model, which posits that the growth of eachspecies is inhibited in a linear fashion by the number of individuals of itsown species and of the other species. This model predicts that stablecoexistence is possible if each species experiences intraspecificcompetition more strongly than it experiences competition from theother species. In epidemiology, similar models can track the prevalence

Fig. 1. (A) structure of the model in Eq. (1). Hosts may become infected by one or twostrains (whichmay be the same or different), but notmore; second infections occur at areduced rate compared to primary infections. Among hosts who have two strainsalready, it is possible (if cN0) for one strain to “knock out” another. See text for moredetails. (B) General approach for a model with multiple infections. The figure showscompartments with different multiplicities of infection, starting from an uninfectedshaded state. Infection from this state leads to an infected state with strain i. Super-infection leads to a dually infected state, a triply infected state, and so on, leadingpotentially but not necessarily to increased infectiousness. The ‘knock-out’ process isnot shown explicitly. If the ‘ladder’ of superinfection is truncated to two co-infections,this model is equivalent to the model of A.

Fig. 2. Evolution of the model in Eq. (1) under various starting conditions. Left ordinateshows the state variables; right ordinate shows f1 (in black). Colors: I1: red; I2: green;I11: blue; I12: tan; I22: magenta. Parameters: c=0.14; k=0.4; q=0.5; β=2; u=1.

3M. Lipsitch et al. / Epidemics 1 (2009) 2–13

of infection with each of two or more strains, and similar qualitativefindings may apply, though often the “resource” – susceptible hosts – isexplicit in epidemiologic models while being implicit in the classicLotka–Volterra competition model. More generally, just as simpleecological models with a single resource tend to predict competitiveexclusion by the species that can maintain equilibrium density at thelowest level of resource (Tilman et al., 1997), simple epidemiological

models with a single host type tend to predict competitive exclusion bythe strainwith the largest basic reproductive number R0 (Anderson andMay, 1991); indeed these two findings are equivalent for the simplesttransmission models since the minimum number of susceptible hoststhat can support equilibrium of a particular strain is inverselyproportional to R0 (Anderson and May, 1991). On the other hand, sucha resource may support stable coexistence if the competitors facetradeoffs between different modes of reproduction. Examples includetradeoffs betweenusinganunused resource anddisplacinga competitor(Levin and Pimentel, 1981), between different modes of transmission(Lipsitch et al., 1996) or between using the primary resource and usingmetabolites of that resource (Levin, 1972).

Long-term coexistence is a problem only if one assumes that strainscompete with one another. Ecological, host species, or other barriersmay in some cases prevent or minimize such interaction, making thedifferent strains essentially independent of each other, such thatcoexistence needs no further explanation. In some virus species, suchas rabiesvirus, “coexistence” may be maintained by circulation indifferent host species or in different, isolated geographic areas, such that

4 M. Lipsitch et al. / Epidemics 1 (2009) 2–13

the prevalence of one strain has no impact on the prevalence of others(Holmes, 2004). In other systems, however, the evidence for competi-tion is compelling. In Streptococcus pneumoniae, for example, the use of avaccine against seven serotypes has resulted in significant absoluteincreases in the prevalence of carriage of serotypes not included in thevaccine, such that total carriage has not changed significantly, but theserotype composition of carriage has been drastically altered (Millar etal., 2006; Huang et al., 2007). These findings are in accord with priorfindings from randomized trials (Obaro et al., 1996; Mbelle et al., 1999;Lipsitch, 2001a,b; Dagan et al., 2002) and with animal experiments(Lipsitch et al., 2000a,b; Dawid et al., 2007) that indicate competitionbetween strains in vivo. Another line of evidence for within-hostcompetition in pneumococci (Feikin et al., 2000) as well as othercolonizing bacteria (Scanvic-Hameg et al., 2002) is the rapid emergenceof large populations of drug-resistant bacteria during antimicrobialtreatment, which suggests that the resistant subgroup was presentbefore treatment but held in check by the larger drug-sensitivepopulation. In influenza, the appearance of new variants by antigenicdrift or, in pandemics, by antigenic shift, is often associated with thedisappearance of prior variants (Grenfell et al., 2004), though thisprocess may take several years in the case of antigenic drift (Rambaut etal., 2008). Strikingly, host mortality (Rohani et al., 2003) or disease-associated behaviors (Rohani et al., 1998) may induce competitiveinteractions even between unrelated pathogens, such as measles andpertussis, thoughwe do not consider such competitive interactions here.

The work described in this paper was stimulated by our efforts tounderstand the apparent long-term coexistence of drug-sensitive anddrug-resistant strains of S. pneumoniae. In attempting to model thisphenomenon, we first considered two parsimonious models, (a) amodel inwhichhostsmaybe colonizedwithonlyone strain or the other,but not both, (Lipsitch, 2001a,b), and (b) a model in which dualcolonization is possible, similar in structure to that described in(Lipsitch, 1997). Model (a) was unsatisfactory because it predictscompetitive exclusionof either resistant or sensitive strains, hence is notcapable of explaining stable coexistence of strains. Model (b) suffersfrom a different problem: it predicts stable coexistence of two strains insome circumstances, and in particular predicts a stable equilibriumfrequency distribution of 50%–50% for two indistinguishable strains.This prediction is not plausible, because it implies that if oneparameterized such a model with no antibiotic use and no differencebetween resistant and sensitive strains, and started resistant strains asonly 1% of the population, they would rise to comprise 50%. A sensiblemodel to understand the interaction between sensitive and resistantstrains would not make the frequency of the resistant strain increasedramatically in the absence of any explicit mechanism promoting itsrise. In the Discussion, we mention another problem for which suchmodels may be relevant in S. pneumoniae, the problem of serotypecoexistence.

If one wants to understand the maintenance of distinct strainswithin in a particular biological system in which competition is likelyand neutral drift alone cannot account for coexistence, then there is aneed to examine other potential mechanisms that may account forsuch coexistence. Biological knowledge in such cases will identifypotential candidate mechanisms and inform model structure: forexample current infection or colonization with one strain maypartially or completely prevent acquisition of a second strain, or ahistory of exposure to one strain (now cleared) may affect the risk ofsubsequent infections with the same or other strains through acquiredimmunity. To evaluate such candidates for their ability to explaincoexistence, it is crucial that any features of the model that affect thelikelihood of stable coexistence are explicit. Put another way, whenthe model is stripped of specific mechanisms that may promote stablecoexistence, it should not produce stable coexistence “for free,” as aside-effect of some implicit assumption in the model structure. Thework in this paper represents an effort to formalize this intuition andidentify conditions underwhich amodel is a sensible starting point for

understanding strain interactions. The effort to understand thecoexistence of resistant and sensitive pneumococci, which builds onthe theory developed here, is deferred to a subsequent paper.

Because detailed data on strain interactions within an individualare often difficult to obtain for ethical and practical reasons, there maybe a number of models consistent with existing data.We can constrainthe set of candidate model structures by requiring that if two strainsthat are functionally indistinguishable (for example, they differ by aselectively neutral single nucleotide polymorphism), the modelshould treat them as such. Specifically, when applied to indistinguish-able strains, a neutral null model for stable coexistence of strainsshould fulfill two criteria: (1, “ecological neutrality”) the dynamics ofthe ecological variables – the number of uninfected hosts and thenumber infected with 0,1,2,… strains – should depend on theecological state variables but, given these, should be independent ofthe identity of the particular strains involved; and (2, “populationgenetic neutrality”) there should be no stable equilibrium frequencyof the strains in the model; rather, it should be possible to chooseinitial conditions to guarantee an arbitrary frequency of strains thatremains constant for all time t≥0. A neutral null model is one that hasno intrinsic mechanism that promotes stable coexistence of indis-tinguishable strains. Throughout this paper, when we use the term“stable coexistence,”wemean that the model has an attracting steadystatewith a fixed proportion of various strains; we specifically excludeneutral stability from our definition of stable coexistence.

In this paper, we define a single set of conditions that guarantees amodel will meet both criterion (1) and criterion (2). Specifically, anyclosed clonal transmission model that can be rewritten in a preciselyspecified sense which meets criterion 1 will itself meet both criterion 1and criterion 2. We call this way of rewriting the model an “ancestor-tracing ecological model” and describe themeaning of this term in detail.In Section 1 we present one model that meets both criteria, show that itmeets criterion (1) and argue informally that it meets criterion (2) inorder to provide some intuition for the structure of the more formalargument, which is given in Section 2. In Section 3 we discuss how thesecriteria relate to previously analyzed models. Up to this point, thediscussion is framed in terms ofmodelswithout strain-specific immunityand with state variables that reflect only the number and identities ofstrains in a given host. In Section 4 we discuss how the considerationsdescribed here generalize to other types of models, including those withmore complex state variables, with strain-specific immunity, and withmutation or migration. We conclude with a discussion of how suchmodels should be used to understand the mechanisms of coexistence.

A transmission dynamic model for two strains, illustrating howboth criteria are met

Consider the model for transmission of two strains shown in Fig.1A and in the following equations:

dI0dt

=− λ1 + λ2ð ÞI0 + u1I1 + u2I2 + uI11 + uI12 + uI22dI1dt

= λ1I0−k1λ1I1−k2λ2I1−u1I1dI2dt

= λ2I0−k1λ1I2−k2λ2I2−u2I2dI11dt

= k1λ1I1−uI11−2ck2λ2I11 + ck1λ1I12dI12dt

=k1λ1I2+k2λ2I1−uI12+2ck2λ2I11+2ck1λ1I22−ck1λ1I12−ck2λ2I12dI22dt

= k2λ2I2−uI22−2ck1λ1I22 + ck2λ2I12λ1 = β1 I1 + qI12 + 2qI11ð Þλ2 = β2 I2 + qI12 + 2qI22ð Þ

ð1Þ

In this model, there are uninfected individuals (I0), individualsinfected with either strain individually (I1 and I2), and individuals

5M. Lipsitch et al. / Epidemics 1 (2009) 2–13

infected with both (I12). In addition, there are now individuals duallyinfected with the same strain (I11 and I22). The force of infection λ1 is(consistent with traditional usage) the hazard for an uninfectedperson to become infected with strain 1, and similarly for strain 2. Foreach strain j=1, 2, the force of infection is equal to an infectiousnessrate constant βj times a weighted sum of the numbers of infectedpersons in each category containing strain j. Dually infectedindividuals are q times as infectious with each of their strains as singlyinfected individuals, hence the I12 individuals contribute β1qI12 to theforce of infectionwith strain 1, and the I11 individuals contribute twiceas much as I12 individuals per capita, 2β1qI11, because they have two“copies” of strain 1. Singly infected individuals may become infectedby a second strain j at a rate equal to kλj, making them become duallycolonized. Dually colonized persons may have one strain “knockedout” by another at a rate c times that for secondary infection ofindividuals singly colonized. Note that this “knocking out”may or maynot be biologically realistic in particular cases, and can be eliminatedfrom the model (setting c=0).

The model of Eq. (1) is intended simply as one example of apossible null model for two strains (and one that we will show isneutral in the senses that interest us), and is not put forth as the bestmodel for all (or even any) pathogen. The assumption of a maximumof two strains is made for analytical simplicity, but clearly one couldimagine more elaborate models with larger numbers of maximumstrains (or perhaps even with no limit on the number of strains) in ahost. Such a model is illustrated in Fig. 1B, which shows thatstructurally the model for an arbitrary number of strains andmultiplicity of infections can be viewed rather simply.

The construction of Fig. 1B also sheds some light on the interpreta-tion dually infected I11 and I22 states. These could correspond to thepresence of different patches on a single host that can be colonized as aresult of multiple infection. Theymust at least correspond to some statewhich is equally reached by re-exposure to any of the strains.

For indistinguishable strains, we set all parameters (ki, ui, βi) equalfor i=1, 2 and set the duration for dual carriage equal to that for singlecarriage, u=u1=u2. (Note: a variant model formulation would be toallow each strain in an infected host to have clearance rate equal to thatin a singly infected host, hence the total clearance rate from dualinfections is twice that from single infections, but only one strainbecomes cleared, e.g. I12 hosts clear to become I1 or I2 hosts, each at rateu.Wedonot consider this further though it alsomeets ourcriteria). Thenit is possible towrite the ecological dynamics of thismodel (dynamics ofthe number of individuals infected with 0, 1 or 2 strains:N0= I0,N1= I1+ I2, N2= I11+ I22+ I12) in terms only of the state variables N0, N1, N2

without reference to the strains involved. In particular,

dN0

dt=−β N1 + 2qN2ð ÞN0 + u N1 +N2ð Þ

dN1

dt= β N1 + 2qN2ð ÞN0−kβ N1 + 2qN2ð ÞN1−uN1

dN2

dt= kβ N1 + 2qN2ð ÞN1−uN2

ð2Þ

Hence this model meets our first, “ecological” criterion. Fig. 2shows how it may meet the second, “population genetic” criterion. InFig. 2A, we initialize the model with I0(0)=0.8, I1(0)=0.19, I2(0)=0.01, I11(0)= I12(0)= I22(0)=0. Fig. 2A tracks the value of the sixIX state variables, and Fig. 2B tracks the proportion of all infectionsthat are strain 1, defined as f1 =

I1 + 2qI11 + qI12I1 + I2 + 2q I11 + I12 + I22ð Þ =

I1 + 2qI11 + qI12N1 + 2qN2

.This is a natural definition — weighting dually infected personsaccording to their relative infectiousness q, though as we will seebelow this choice need not be made. Clearly, as the state variableschange, the value of f1 remains constant at its starting value, in thiscase 0.95. A similar result occurs in Fig. 2B, where we start with onlydually infected individuals, here starting with strain 1 at a frequencyf1=0.1: I0(0)=0.8, I1(0)= I2 (0)=0, I11(0)=0.02, I12(0)=0, I22(0)=0.18. Here, again, the frequency of strain 1 stays constant as thesystem evolves. If, on the other hand, we start with both singly and

dually infected hosts (I0(0)=0.60, I1(0)=0.19, I2(0)=0.01, I11(0)=0.02, I12(0)=0, I22(0)=0.18) with different frequencies of strain 1,we find that the value of f1 does not remain constant with time, butchanges modestly as the singly and dually infected individuals changetheir relative contributions to the overall strain composition (Fig. 2C).

These numerical results suggest that it is possible to start themodelwith an arbitrary frequency of strain 1 at time t=0 and maintain thatfrequency for all tN0 as the model evolves. This strictly constant valueof f1 appears to be possiblewhen the starting conditions involve singlyor dually infected hosts, but not when both singly and dually infectedhosts are present at the start, with different starting frequencies of thestrains. However, the modest change in f1 with time when both typesof hosts are present at the start suggests that somehow even in thatcase, the final state of themodel depends on the initial conditions, andthis is borne outwhen other combinations of strain frequencies in dualand singly infected hosts are chosen (results not shown).

In summary, we have constructed a model whose ecologicaldynamics are independent of strain identity, and we have showninformally that starting conditions in this model can be chosen toguarantee an arbitrary frequency of strain 1 that remains constant in themodel over time. In thenext sectionweuse the intuitiongained fromthisexample to motivate a general proof that if the “ancestor-tracing”ecological dynamics of a closed clonal transmission model areindependent of strain identity, it is always possible to choose initialconditions thatwill guarantee anydesired frequencyof the strains, at anytime. This implies that any combination of frequencies of strains can be along-termequilibriumof themodel, and thus that themodel, by virtueofits structure, does not encode for any selection on these strains. We willalso show the converse, namely that some commonly used models onlypermit equilibria where all the strains are at equal frequencies, and thusimplicitly encode for balancing selection acting on the strains.

General conditions that assure a transmission model meets thecriteria for a neutral null model

Informal overview of this section

This section identifies a set of conditions for a transmission modelthat guarantee that the two criteria for a neutral null model — (a)ecological dynamics independent of strain identities and (b) no stableequilibrium of frequencies, but rather the ability to choose initialconditions that will maintain any arbitrary strain frequencies through-out the time evolution of the model. Although there is much notationinvolved, the basic concept of this section is that both conditions aremetwhenone can followaprocedure like that in Section1, startingwithonlyuninfected and (say) singly infected hosts and initializing strainfrequencies among the singly infected hosts at time 0 to the desiredfrequencies. This in turn is possible when one can trace every strain atany time t back to a unique ancestor of the same type present at time 0,in a fashion that never involves terms identifying particular strain types.If this is possible, then one can start with singly infected hosts only, anddescendants of the strains infecting these hosts, by symmetry, will havethe same strain compositions as the starting population. Since all strainsin themodel will be descended from these hosts, all strains in themodelwill have the desired frequencies of strain types.

The argument involves introducing a scheme for tracking not onlythe ecological dynamics, but the “ancestor-tracing ecologicaldynamics,” meaning state variables track not only the multiplicity ofthe host carrying a given strain, but also the multiplicity of the host ofa strain’s ancestor. These are the variables called M•.

The argument is generalized slightly by following not only hostmultiplicity, but also host “classes,” denoted by θ —which may refer togender, age, antibiotic treatment or other fixed or time-varyingcharacteristics. This becomes useful later but merely adds to thenotation in the proof. A first-time reader is advised to ignore all the θ inthis section, equivalent to assuming only a single class of hosts.

6 M. Lipsitch et al. / Epidemics 1 (2009) 2–13

Definitions and notation

We focus on a a dynamical system dIθxdt = gθx

YI; t

� �; x=0 N X;

θ=1 N Θ, in which the state variables Iθx can be understood asreferring to the number of hosts in a population of class θ withcolonization status x, x=1…X. The “class” of a hostmay be understoodas a vector of host covariates (possibly including time-varying orconstant characteristics such as age, sex, antibiotic treatment state,etc.), while the colonization status indicates which strains, at whichmultiplicity, currently infect the host. Let every host be capable ofbeing infected by at most n strains simultaneously, and let there be Ystrains in circulation. Let the x index all possible ordered listsYh

x= hx1;h

x2; :::h

xn

� �, where Yh

xis a nondecreasing list of the strain

identities of the strains infecting a host. Someof the entrieswill be zeroin hosts not infected with the full number of strains, and some of theentries may be repeated if hosts are infected more than once with agiven strain. The requirement that the list is nondecreasing defines aunique order for the strains infecting a given host.

Define the multiplicity of state x, or m(x), as the number of strainsinfecting a host in state x.

For any dynamical system of the {Iθx} we can define the ecologicalstate variables Nθi = ∑

m xð Þ = iIθx as the number of hosts of class θ with

multiplicity i. Thus the ecological state of the system, defined byYN tð Þ= Nθi tð Þf g tells us howmany hosts of each class exist infected by0, 1, 2,…, n strains, without reference to which strains they areinfected by.

In what follows we will want to trace the “ancestry” of strainsinfecting various hosts. In particular, for some dynamical systems itmakes sense to define every strain infecting a host as descended from astrain that was present at time 0, which we call the ancestor. We callsuch a model a closed clonal transmission model. It is closed in thesense that no infecting strain can enter the model (though uninfectedhosts may enter, and infected or uninfected hosts may die). It is aclonal transmissionmodel in that all strains were either present at thestart or result from transmission, and in such a model, transmissionimplies clonal descent from a unique ancestor.

Table 1Corresponding terms in ecological and ancestor-tracing ecological models

Event Term in equation for N:θi; iz1 Event

1. Host demographics none Birth, death, etc.2. Change in host covariatesfrom θ to θ′

−χθθ0 Nθið Þ Aging, acquisition of immu

(corresponding, positive terms for entry from other classe

3. Clearance:a. Clearance of one or morestrains from a [θi] host

−uθi Nθið Þ a. Clearance of h strains, rewho harbors one or more

(corresponding, positive terms for entry from other multipto the below, which keeps track of the changes in multipli

b. Clearance of h strainsfrom a [θ (i+h)] host

uθ i + hð Þ Nθ i + hð Þ� �

b. Clearance of h strains, n[θ(i+h)] host who also hafrom a θ′j ancestor

4. New infections:a. Infection of a [θ(i−1)] hostby a new strain, which mayoriginate from any hostclass index by θ⁎, d

∑θ4;d

Λ θ i−1ð Þ½ �← θ4j½ �ðNθ i + hð Þ;YNÞ1 a. Infection of a [θ(i−1)] h

b. Infection of a [θ(i−1)] hfrom a [θ′j] ancestor by an

(corresponding negative terms for loss from lower multipl

NOTE: The force of infection of strains descended from [θ′, j] ancestors, λ[θ′, j], would be defi

the β[θi] are transmission coefficients per strain infecting a [θi] host. The total force of infect1 Here Λ θ i−1ð Þ½ �p θ4d½ � Nθ i−1ð Þ;

YN

� �is the force of infection exerted on hosts of class θ, i−1 by

general, this force of infection may be a complicated function of the number of θ, i−1 hostsmodel like the one in Eq. (1), the term is much simpler, i.e. Λ θ i−1ð Þ½ �p θ4d½ � Nθ i−1ð Þ;

YN

� �= βθ

Λ θ i−1ð Þ½ �p 11½ � Nθ i−1ð Þ;YN

� �; N Λ θ i−1ð Þ½ �p ΘY½ � Nθ i−1ð Þ;

YN

� �n ois a list of the forces of infection exerte

tracing model apportions this force of infection according to the ancestry of the strains curr

For example, a classic SIR — type transmission model is a closedclonal transmission model, since every infection traces back through achain of infection events to an initial infective. A model withimmigration of infectives, on the other hand, would not be closedbecause some strainswould appear with no ancestor at time 0. Amodelwhere descent cannot be traced clonally (e.g. where strains appear byrecombination from two ancestors) would not strictly meet ourdefinition of a clonal transmission model, since no unique ancestorcan be identified. It is likely that recombination could be readilyincorporated intonullmodels of the sortweare consideringhere, butweignore this issue because of the notational difficulties it would create.

We define the ancestor-tracing ecological state variables M[θi]←[θ′k] (t)to be the number of strains infecting individuals of class θ infected withmultiplicity i at time t whose ancestor at time 0 was in an individual ofclass θ′ infected with multiplicity k. In a closed model, note that M[-

]←[θ′k](0)= iNθi(0) if i=k and θ =θ′ and 0 otherwise. Moreover,

∑Θ

θ ′= 1∑n

k = 1M θi½ �p θ ′k½ � tð Þ=iNθi tð Þ ð3Þ

We define the ancestor frequencies μ θi½ �p θ ′k½ � tð Þ= M θi½ �p θ ′k½ � tð ÞiNθi tð Þ as the

frequency of strains descended from θ′, k ancestors among all thosestrains currently infecting θ, i hosts. We will use the vector of suchfrequencies Yμ θi½ � tð Þ= fμ θi½ �p 1;1½ �; N μ θi½ �p Θ;Y½ �g which represents thefrequencies of all possible ancestries among strains currently infectingθ, i hosts; the sum of the components in each vector Yμ θi½ � tð Þ is 1 bydefinition.

For models broadly of the form of traditional SIR models, everyterm appearing in an equation in the full model reflects one of thefollowing four types of events:

1. Changes in the demographics of uninfected hosts (births, deaths,etc.)

2. Changes in the covariates of individual hosts, in a way that maydepend on their infection status (e.g. aging, acquiring immunity,etc.)

Term in equation for M:½θi�←½θ0 j�

nonenity, etc., exit from class θ −μ θi½ �← θ ′j½ �χθθ ′ Nθið Þ

s into class θ′)

gardless of ancestry, from a [θi] hoststrains descended from a [θ′j] ancestor

−μ θi½ �← θ ′j½ �uθi Nθið Þ

licities into multiplicity i, in additioncity for the strains not cleared)

ot descended from a [θ′j] host, from arbors one or more strains descended

μ θ i + hð Þ½ �p θ ′j½ �uθ i + hð Þ Nθ i + hð Þ� �

ost by a strain descended from a [θ′j] ancestor Yμ θjdYΛ θ i−1ð ÞðNθ i−1ð Þ;

YNÞ1

ost already carrying a strain descendedy strain

μ θ i−1ð Þ½ �p θ ′j½ � ∑θ4;d

Λ θ i−1ð Þ½ �; θ4d½ �ðNθ i−1ð Þ;YNÞ

icities)

ned in a “classic” transmission model tracing ancestry as λ θ ′j½ � = ∑θi½ �β θi½ �M θi½ �p θ ′j½ � , where

ion Λ = ∑θ;kλ θk½ � is the sum of all forces of infection from strains of all ancestries.

hosts of class θ⁎, d, by which they acquire an additional strain and move into class θ, i. In(the infectees) and the number of all other types of hosts. In an ordinary transmission

dNθ i−1ð ÞNθd , where βθ1=β, βθ2=2qβ. The corresponding vectorYΛ θ i−1ð Þ Nθ i−1ð Þ;

YN

� �=

don θ, i−1 hosts by all types of hosts. The dot product in the equations for the ancestor-ently infecting each of these types of infecting hosts.

7M. Lipsitch et al. / Epidemics 1 (2009) 2–13

3. Clearance of one or more strains from a host, resulting in areduction in the host’s multiplicity

4. Infection of a host with a new strain, resulting in an increase in itsmultiplicity and no change in the presence of other strains infectingthat host. In this case, for a closed clonal transmission model, thereis a unique host (the infectee)-who becomes infected with aparticular strain that is of the same type as a strain in a unique host(the infector).

In addition, a single event may encompass more than one of thesechanges. For example, in model (1), the terms including c refer toevents that combine infectionwith a new strain and loss of an existingstrain. Table 1 shows in general terms, using general functional formsfor these events, how such terms appear in both the ecological andancestor-tracing ecological models corresponding to a particular fullmodel.

As an example of the ancestor-tracing ecological dynamics, weconsider the model of Eq. (1) and Fig. 1 (where there is only one classof host, so we ignore θ and θ′ and drop their subscripts). Here, we have

M:11 = β M11 + qM21ð ÞN0−uM11−kλtotalM11

M:12 = β M12 + qM22ð ÞN0−uM12−kλtotalM12

M:21 = βk M11 + qM21ð ÞN1 + kλtotalM11−uM12

−cβk M12 + qM22ð ÞM21 + cβk M11 + qM21ð ÞM22

M:22 = βk M12 + qM22ð ÞN1 + kλtotalM12−uM22

+ cβk M12 + qM22ð ÞM21−cβk M11 + qM21ð ÞM22λtotal = β M11 +M12 + qM21 + qM22ð Þ= β N1 + 2qN2ð Þ

Note that M0·=0 since by definition hosts of multiplicity 0 haveno strains infecting them, and M·0=0 since hosts of multiplicity 0 attime 0 are not the ancestors of any infections. Also note that the termsinvolving c, which drop out of the ecological system because they donot change strain multiplicity, reappear in the ancestor-tracingecological system because they change the identity of strains.

We can write the M• in matrix form as M{θ, i}{θ′,k}=M[θi]←[θ′k],where the brackets in subscripts indicate a single number between 1and Θn chosen to indicate a particular combination of host class andmultiplicity. Then M(t) is a square matrix of dimension Θn×Θn, witheach row representing one of the Θn possible combinations of strainmultiplicity and host class (for strains at time t), and each columnrepresenting the same (for their ancestors).

Define Rjx as the number of strains of type j present in anindividual of state x, that is, the number of entries equal to j in the listof {h1x, h2x,…hn

x}.Define Sθij(t) as the number of strain j infections in hosts of class θ

infected with multiplicity i at time t. Formally, we defineSθij tð Þ= ∑

m xð Þ = iRjxIθx tð Þ. It follows that

∑Y

k = 1Sθik tð Þ= iNθi tð Þ: ð4Þ

Define a strain frequency for strain j in the population as the totalrepresentation of strain j among hosts with all multiplicities, wherehosts of class θ with multiplicity i are given arbitrary weight wθi inboth numerator and denominator, such that

fj tð Þ=∑Θ

θ = 1∑n

i = 1wθiSθij tð Þ

∑Θ

θ = 1∑Y

y = 1∑n

i = 1wθiSθiy tð Þ

: ð5Þ

Define

/θij tð Þ= Sθij tð Þ

∑Y

k = 1Sθij tð Þ

=Sθij tð ÞiNθi tð Þ ð6Þ

as the frequency of strain j among hosts of multiplicity i and class θ at

time t. These frequencies add to 1 for each multiplicity i: ∑Y

j = 1/θij =1.

Formal statement of conditions

CLAIM: Consider a dynamical system of {Iθx(t)} which represents aclosed clonal transmission model. Suppose that the ancestor-tracingdynamical system associated with this model for the case ofindistinguishable strains, using variables {M[θi]←[θ′k](t)}, can bewritten in a form independent of strain identities:

M:½�i�←½�0k� tð Þ= F θi½ �p θ ′k½ � t;N0 tð Þ;M tð Þð Þ for all θ; θ ′a 1;Θ½ �; i; ka 0;n½ �: ð7Þ

Then the two conditions for a neutral null model are met. (a) Theecological dynamical system YN tð Þ= Nθi tð Þf g is independent of strainidentities; and (b) there is no locally asymptotically stable equilibriumstrain frequency in the model, and it is possible to devise startingconditions YI 0ð Þ to produce a fixed vector of strain frequenciesYf tð Þ=Yf 0ð Þ for all time tN0.

PROOF: Condition (a) follows trivially. Each state variable in theecological model is proportional to a sum of variables in the ancestor-

tracing ecological model, ∑Θ

θ ′= 1∑n

k = 1M θi½ �p θ ′k½ � tð Þ= iNθi tð Þ. Thus if the

ancestor-tracing ecological system is independent of strain frequen-cies, so is the ecological system. The converse is not true: it is possibleto have a model with a strain-independent ecological system whoseancestor-tracing ecological system is not strain-independent. Forexample, a model similar to that in Fig. 1 and Eq. (1) but lacking thetransitions from dual colonization with both strains to dual coloniza-tion with a single strain, the terms containing ckλiI12, i=1, 2, wouldhave the same ecological dynamics as Eq. (1), since these transitionsdo not change the multiplicity of strains in any host, but would nolonger have strain-independent dynamics for their ancestor-tracingecological dynamics, and indeed would for some parameters yieldstable coexistence of indistinguishable strains.

To prove condition (b), the key point is to trace back every strainpresent in a host at time t to its ancestor, and then use the fact that ithas the same type as its ancestor to calculate the total number ofstrains at time t that are of any given type j. From this total number wecan calculate strain frequencies.

The tracing back of all strains is possible because we areconsidering a closed clonal transmission model, for which Eq. (3)holds, assigning every strain a unique ancestor.

The fact that the ancestor-tracing ecological dynamics can bewrittenwithout reference to strain frequencies implies that per capita,any ancestral strain infecting a [θ′k] host will leave the same numberof descendants as any other ancestral strain in a [θ′k] host, regardlessof which strain type these hosts carry. Combining this with theassumption that the system is clonal (so the descendants are of thesame type as the ancestor), it follows that the descendants of ancestralstrains infecting [θ′k] hosts will have the same frequencies of straintypes as the ancestors did. Putting these assumptions together tomake the key step in the proof, we have:

Sθij tð Þ= ∑Θ

θ ′= 1∑n

k = 1M θi½ �p θ ′k½ � tð Þ/θ ′kj 0ð Þ: ð8Þ

Eq. (8) states that at any time t, we can calculate the number ofstrain j infections in hosts of multiplicity i and type θ by tracing backthe source of all strains in hosts of multiplicity i and type θ to theirorigins in hosts of multiplicity k and type θ′ at the start of the model,through ancestor-tracing ecological calculations (with the M•s) andthen weighting the contribution from such [kθ′] hosts by therepresentation of strain j in those original hosts.

Now, for any t and any desired choice of strain frequencieswf j tð Þ,

we can choose initial conditions for the full model that will producewf j tð Þ. In particular, we can infect only hosts of a single host class (callit class θ=1) and choose initial conditions such that only hosts ofmultiplicity 0 (uninfected) and 1 (singly infected) are present at time0, and such that a proportion of infected hosts fj 0ð Þ=w

f j tð Þ are

8 M. Lipsitch et al. / Epidemics 1 (2009) 2–13

infected with strain j. Then because there were no infections ofmultiplicity N1 present at the start, we have Sθij(0)=0 if (iN1 orθN1), and Mθi←11(t)= iNθi(t), and Mθi←θ′k(t)=0 if (kN1 or θN1).Then at time t we will have

fj tð Þ=∑Θ

θ = 1∑n

i = 1wθiSθij tð Þ

∑Θ

θ = 1∑Y

y = 1∑n

i = 1wθiSθiy tð Þ

=∑Θ

θ;θ ′= 1∑n

i = 1∑n

k = 1wθiM θi½ �p θ ′k½ � tð Þ/θ ′kj 0ð Þ

∑Θ

θ;θ ′= 1∑Y

y = 1∑n

i = 1∑n

k = 1wθiM θi½ �p θ ′k½ � tð Þ/θ ′ky 0ð Þ

=∑Θ

θ = 1∑n

i = 1wθiM θi½ �p 11½ � tð Þ/11j 0ð Þ

∑Θ

θ = 1∑Y

y = 1∑n

i = 1wθiM θi½ �p 11½ � tð Þ/11y 0ð Þ

=/11j 0ð Þ ∑

Θ

θ = 1∑n

i = 1wθiM θi½ �p 11½ � tð Þ

∑Θ

θ = 1∑n

i = 1wθiM θi½ �p 11½ � tð Þ

=/11j 0ð Þ= fwj tð Þ

ð10Þ

Here, the first equality is Eq. (5), the second follows from Eq. (8),the third from the starting conditions, and the fourth by rearrange-ment. Thus we have constructed a way to obtain any desired strainfrequency for all tN0 by choosing the correct initial conditions. Thisimplies that there is no stable equilibrium for the strain frequencies,QED.

Practicalities: how can one tell if a model meets the conditions for aneutral null model?

In principle, it may be tedious to write out the full ancestor-tracingecological system {M[θi]←[θ′k](t)} associatedwith a transmissionmodellarger than the simple system used for illustration in Section 2.1. It iseasier, however, to calculate the ecological system without ancestor-tracing, {Nθi}, simply by adding up the differential equations forall terms with a given multiplicity according to the definitionNθi =∑m xð Þ = iIθx, and substituting appropriately. If all strain-specificterms do not drop out of this formulation, then clearly the ecologicalcriterion is violated, which implies that the conditions of Section 2.2are violated, and the model is not a neutral null model. This strategy isfollowed in the next section, in Eqs. (12) and (13).

If the ecological system can be written without terms that identifyparticular strains, we still need to know whether the ancestor-tracingecological system can bewritten in such away. As noted in Section 2.2,a strain-independent ecological system may be associated with anancestor-tracing system that cannot eliminate references to particularstrains. Hence, if the ecological criterion is satisfied, it becomesnecessary to write down and inspect the ancestor-tracing ecologicalsystem to confirm that the population genetic criterion is satisfied.Since this involves bookkeeping and arithmetic, rather than thesometimes complicated algebra of stability analysis, we suspect thatthis is easier in most real cases than performing the formal stabilityanalysis directly.

Starting from mixed initial states

For proving our result, it is sufficient to show that any equilibriumcan be reached. Eq. (10) shows that this can be done by starting from a‘pure’ initial conditions with all infections in the singly infected (i=1)first state (θ=1) inwhich case the strain frequencies stay constant forall time tN0. This can trivially be generalized to any other ‘pure’ initialstates (θi), but as we saw in simulations in Section 1, starting from a

mixed initial state will in general result in a transient initial change instrain frequencies before an equilibrium is reached.

In general, it can be shown by contradiction that the strainfrequencies are equal for all the ecological states in the finalequilibrium state, i.e.

limtY∞/θij tð Þ= f ∞j ;

as long as all the host types are “connected” by transmission, perhapsindirectly.

In this case the matrix M becomes degenerate at the equilibrium,and the final frequencies are given by

f ∞j =limtY∞ ∑

Θ

θ ′= 1∑n

k = 1M θi½ �p θ ′k½ � tð Þ/θ ′kj 0ð Þ

limtY∞iNθi tð Þfor any choice of i and θ. The matrix

K=limtY∞M θi½ �p θ ′k½ � tð Þ

limtY∞iNθi tð Þspecifies the transition from the initial strain frequencies to theequilibrium. To understand how the model can simultaneously haveany combination of equilibrium strain frequencies and still havedependence on initial conditions, one can draw an analogy with theHardy–Weinbergmodel. Starting from a diploid state, a full generationof replication is required before the Hardy–Weinberg equilibrium isobtained. In this case, an equivalence can be drawn where themultiplicity of infection i is a generalization of polyploidy for infectionstates, and K is the next generation operator for strain frequencies.

Relationship to prior models

For pathogens without acquired immunity, two simple sorts ofmodels have been used in the past to model strain competition andcoexistence. The first simply assumes that a host may be infected at agiven time with one strain or another, but not both, and onlyuninfected hosts may become infected. This model, shown in Fig. 3A,may be written as follows:

dI0dt

=− λ1 + λ2ð ÞI0 + u1I1 + u2I2dI1dt

= λ1I0−u1I1dI2dt

= λ2I0−u2I2λ1 = β1I1λ2 = β2I2

ð11Þ

and is a special case of mode1 1, with k=0. As such, it by definitionmeets our criteria, and if the strains are indistinguishable (β1=β2,u1=u2), then the relative frequency of each strain will remain at itsinitial value indefinitely.

The simplest model that permits co-infection of a single host(Dietz, 1979; Gupta et al., 1994; Lipsitch, 1997) is shown in Fig. 3B,with equations

dI0dt

=− λ1 + λ2ð ÞI0 + u1I1 + u2I2dI1dt

= λ1I0−u1I1−k2λ2I1 + u2I12dI2dt

= λ2I0−u2I2−k1λ1I2 + u1I12dI12dt

= k2λ2I1 + k1λ1I2− u1 + u2ð ÞI12λ1 = β1 I1 + qI12ð Þλ2 = β2 I2 + qI12ð Þ

ð12Þ

As previously analyzed, with q=1 (so dually infected hosts areequally infectious with each serotype as singly infected hosts) andk1=k2=kN0 (so dual infection is possible in both directions), this

Fig. 3. Alternative model structures. (A) Model equations 11; (B) Model equations 12;(C) Model equations 14.

9M. Lipsitch et al. / Epidemics 1 (2009) 2–13

model fails our criteria for an appropriate null model that does not“build in” stable coexistence. The ecological dynamics of this modelcannot be written independent of strain composition, since here,

dN0

dt=−β N1 + qN2ð ÞN0 + uN1

dN1

dt= β N1 + qN2ð ÞN0−uN1−2kβ I1I2 + qN2N1½ �

dN2

dt=−2uN2 + 2kβ I1I2 + qN2N1½ �

ð13Þ

Moreover, this model has a single stable equilibrium includingboth strains whenever the strains are indistinguishable and secondaryinfection is possible (kN0) (Dietz, 1979; Gupta et al., 1994; Lipsitch,1997). Before considering how we may adjust this model to meet ourconditions, it is worthwhile to consider why it violates them.

Note that in this model, an individual infectious with one strainmay become more infectious in total (since q=1 makes duallyinfected hosts twice as infectious as singly infected ones) only if heencounters the other strain, not if he encounters the same strain again.This means that, for a given prevalence of single infection, each strainhas more opportunities for transmission as the proportion of singlyinfected hosts with the other strain increases. In this way, each strainpromotes the other’s success, tending to promote stable coexistence.This can be seen in the term including I1I2 for new infections in Eq.(13).

Another equivalent way of stating this is that, in this model unlikethe earlier model of Eq. (1), superinfection leading from a pure I1 or I2state to a mixed I12 state only arises due to within-host balancingcompetition between the two strains. The model of Eq. (12) encodeswithin-host balancing selection, and as a result the predicted straindynamics show population-level balancing selection.

It is important to note at this point that the modeler has a choiceabout how to apply a given model, such as Model 13, to indistinguish-able strains. In the discussion so far, we have assumed that thesecondary infection coefficients k1 and k2 are fixed constants, whichare nonzerowhether or not the strains are different (because, as notedabove, secondary infection arises due to balancing selection within ahost). Suppose instead that secondary infection could only occur if theincoming strain was more “fit” in some sense (e.g. had higher in vivogrowth rate) than the resident strain. For simplicity, let each strainhave some defined level of within-host competitive ability wi, and letki=k(wi−w∼i), with the restriction that k(v)=0 for v≤0, and k(v)≥0 for vN0. Under this assumption, the equations become asfollows (suppose without loss of generality that strain 1 is always ofgreater or equal fitness than strain 2):

dI0dt

=− λ1 + λ1ð ÞI0 + u1I1 + u2I2dI1dt

= λ1I0−u1I1 + u2I12dI2dt

= λ2I0−u2I2−k1λ1I2 + u1I12dI12dt

= k1λ1I2− u1 + u2ð ÞI12λ1 = β1 I1 + qI12ð Þλ2 = β2 I2 + qI12ð Þ

On their face, these equations do not look neutral, but when theyare applied to indistinguishable strains, neither strain can establishsecondary colonization, hence k1→0 and the model reduces to theneutral Model 11. However, when the two strains are distinguishable,strain 1 may be able to establish secondary colonization. This findingemphasizes that the idea of “applying a model to indistinguishablestrains” depends on one’s assumption of how various parametersbehave when the strains are indistinguishable.

Returning to the assumption that the ki are always nonzero, oneway to modify Model 12 to be neutral when applied to indistinguish-able strains is to include two additional states of “dual” infection with

the same strain, as in model 1. In this way, singly infected hosts canbecome infected with another strain, at equal rates whether it is thesame strain they already have, or not.

A second, slightly simplermodel would be one that keeps almost thestructure of model Eq. (12), but adds the possibility that dually infectedhosts may have one strain “knocked out” by contact with the other

10 M. Lipsitch et al. / Epidemics 1 (2009) 2–13

strain, returning to being singly infected. This occurs at a rate c times asgreat as second infection of a singly infected person. Also, we now allowdually infected hosts to clear both strains simultaneously; this does notchange anything important but allowsus tomakea structure identical tothat in Eq. (1). Themodified system is shown in Fig. 3C and the followingequations, written now with J to facilitate comparisons below:

dJ0dt

=− λ1 + λ2ð ÞJ0 + u1J1 + u2J2 + u3J12dJ1dt

= λ1J0−u1J1−k2λ2J1 + ck1λ1J12dJ2dt

= λ2J0−u2J2−k1λ1J2 + ck2λ2J12dJ12dt

= k2λ2J1 + k1λ1J2 + u3J12−c k1λ1 + k2λ2ð ÞJ12λ1 = β1 J1 + qJ12ð Þλ2 = β2 J2 + qJ12ð Þ

ð14Þ

In this case, it is still not possible to write the ecological dynamicswithout including strain identifiers. However, note that for theparticularparameter values c=q=1/2, Eq. (14) become equivalent to Eq. (1),with J1=I1+ I11 and J2= I2+ I22, hence its ecological and ancestor-tracing ecological dynamics become strain-independent. Whicheverway themodel iswritten, it can be interpreted as follows: an individual iseither infected or not. If infected, he exerts a certain force of infection,regardless of which strains infect him (since q=1/2, dually infectedindividuals are equally infectious in total to singly infected individuals). Ifhe is infected, hemay lose half of his infectiousness to a new strainwithwhichhehas contact, at a ratek times that atwhichanuninfectedpersonbecomes infected given contact. In a two strainmodel, an individualwithboth strains who acquires a strain through subsequent infectionwill halfthe time lose the strain of the same type he acquired, hence c=1/2.

To summarize the implications for models similar to that in Eq.(12): if dual infection in individual hosts is impossible (Eq. (11)), themodel is neutral when applied to indistinguishable strains. If dualinfection is possible (for any pair of strains), and is implemented as itis in Eq. (12), then the model is not neutral, and tends to promotecoexistence of indistinguishable strains. Two fundamentally differentsolutions are possible to make a neutral null model. First, one canretain a model structure like that of Eq. (12), but with the proviso thatindistinguishable strains cannot cause dual infection. This would beplausible if a within-host fitness advantage were required for thesecond strain to growwithin an already colonized host. In this case, nosecondary infections would occur for indistinguishable strains, andonly one of the ki would be positive when two different strains wereconsidered. Second, one can add two more states (dual colonizationwith two of the same strain), transforming Eq. (12) into Eq. (1). This isequivalent, as we showed, to maintaining the states named in Eq. (12)but allowing secondary infections of the dual state, returning hosts tothe singly colonized state, for a particular choice of parameters.

Generalizations

More complex state variables

In the foregoing, we have considered only models in which a host’sstate x is determined by his multiplicity of infection and the identity ofthe strain(s) present. More generally, models might permit a host with,say,multiplicity2, to bemore infectiouswith oneof the strains thanwiththe other. As long as the determining factor for which strain is moreinfectious is not the identity of the strain, this should not violate ourconditions for neutral null models. For example, perhaps the first strainto infect the host is more infectious than the second (or vice-versa), orthe two strains infecting a host are always unequally infectious, but it israndomwhich strain infecting the host is more infectious.

Suchmodelsmaybe easilyaccommodatedwithinour proof, simply byexpanding the possible number of states, such that (for example) hYx isnowa list ordered by the relative infectiousness (or order of arrival) of the

strains infecting thathost (rather than innondecreasingorderof the strainidentities as defined in Section 2). Then, the i and k indices of M[θi]←[θ′k]

must index not multiplicities, but positions within hosts of a particularmultiplicity. For example, if hosts may have up to three strains of unequalinfectiousness, i andkwould range from1 to6, indicating thefirst positionin a singly infected host; the first and second position in a dually infectedhost; and the first, second and third position in a triply infected host.

This is an important point. A model need not be symmetric in allrespects in order to meet our criteria for neutrality. Rather, it must besymmetric with respect to the identities of the strains.

Models with immunity

Models with strain-specific immunity – in which prior experienceof strain jmakes a host less likely to be infectedwith strain j to a degreegreater than it protects against other strains – are well known forpromoting stable coexistence of strains (Gupta et al., 1996; Gupta andMaiden, 2001). This is the standardmechanism of negative frequency-dependent selection often invoked to explain stable antigenic poly-morphism (Lipsitch andO'Hagan, 2007). A simplemodel of that type isshown in Fig. 4A, and is given by the following equations:

dXdt

=− λ1 + λ2ð ÞX+w1Z1 +w2Z2 +wZ

dY1

dt= λ1X−u1Y1

dY2

dt= λ2X−u2Y2

dZ1dt

= u1Y1−kλ2Z1−w1Z1dZ2dt

= u2Y2−kλ1Z2−w2Z2dY1;2

dt= kλ1Z2−u1Y1;2

dY2;1

dt= kλ2Z1−u2Y2;1

dZdt

= u1Y1;2 + u2Y2;1−wZ

λi = βi Yi + qYi;fi� �

; i=1;2

ð15Þ

Such models should fail our test for neutral null models, since theyoften predict stable coexistence of more than one strain. They fail theecological criterion because, by definition, a state of immunity to onestrain affects the transmission of that strain more than the transmis-sions of other strains, hence the ecological dynamics cannot bewritten independent of strain identity. It should be emphasized thatthis does not mean that there is anything intrinsically wrong withmodels including strain-specific immunity, only that the strain-specific immunity itself is a mechanism promoting stable coexistence.

On the other hand, it is possible to have a neutral model thatincludes immunity, as long as the immunity does not depend on theidentity of the strains with which an individual has been infected. Anexample is shown in Fig. 4B and is given by the equations:

dXdt

=− λ1 + λ2ð ÞX+w ′Z ′+wZ

dY1

dt= λ1X−u1Y1

dY2

dt= λ2X−u2Y2

dZdt

= u1Y1 + u2Y2−k λ1 + λ2ð ÞZ−wZ

dY ′1dt

= kλ1Z−u ′1Y ′1dY ′2dt

= kλ2Z−u ′2Y ′2dZ ′dt

= u ′1Y ′1 + u ′2Y ′2−w ′Z ′

λi = βi Yi + qY ′ið Þ; i=1;2

ð16Þ

This fits nicely into the framework developed in Section 2, wherethere are three host classes, “naïve”, “infected once” and “infected

Fig. 4. Models with immunity. (A) A model with strain-specific immunity, Eq. (15). Thismodel is not neutral, and it promotes stable coexistence of the two strains. (B) A neutralmodel with immunity, Eq. (16). This model has immune states that depend only on thenumber of times a host has been infected, but not onwhich strains have infected the host.

11M. Lipsitch et al. / Epidemics 1 (2009) 2–13

twice” (denoted by unprimed and primed variables respectively),though the memory of these prior infections can wane; thus hereθ=3. We can rewrite the dynamics now for identical strains as

dN10

dt=−β N21 + qN31ð ÞN10 +wN20 +w ′N30

dN21

dt= β N21 + qN31ð ÞN10−uN21

dN20

dt= uN21−kβ N21 + qN31ð ÞN20−wN20

dN31

dt= kβ N21 + qN31ð ÞN20−u ′N31

dN30

dt= uN31−w ′N30

ð17Þ

where the Nθi indicate hosts with θ−1 prior infections and i currentinfections. The key here is that we defined the host classes indexed byθ so that they could be time-varying and even depend on the host’shistory, as long as they do not depend on the identity of strains withwhich a host was infected.

In summary, models with immunity that is strain-specific are notneutral because the ecological dynamics depend on the identity ofstrains with which individuals have been previously infected, andbecause they tend in fact to promote stable coexistence of strains.Indeed, suchmodels represent one class of mechanistic models for thestable coexistence of strains. More general models of strain immunityinvolve matrices of cross immunity between strains, and can exhibitcomplex regular or chaotic oscillations in strain frequencies over time(e.g. (Gupta et al., 1998)). Models with immunity that reflectsprevious exposure but is not strain-specific may be neutral, asexemplified by Eq. (15), and these may be understood within theframework of Section 2 by considering a host’s history of exposure asdetermining his class, indexed by θ.

Models with immigration or mutation

Our main result was stated for “closed clonal transmissionmodels,” namely those in which each infection is uniquely descended

from a particular infection of the same type present at time 0. Thisproperty is used in the proof since it implies the existence of theMθi←θ′k necessary for Eq. (7).

Models with immigration of hosts infected with particular strainsviolate the definition of “closed clonal transmission models,” sincesome infections are not descended from infections present at time 0but rather from immigrant infections. Models with mutation of typesviolate this definition because, although a particular infection maydescend from one present at time 0, it was of a different type. Neithertype of model can be accommodated within our framework of neutralnull models. This is understandable, because in a model withimmigration, strain frequencies will reflect to some extent theinfluence of immigration, hence cannot be controlled for all timesby the choice of starting conditions (Hubbell, 2001).

While models which meet our two tests for null neutrality cannotinclude mutation or immigration, they can form the basis for theexploration of these phenomena on strain diversity in a context wherethe effect of strain selection is understood and controlled for.Epidemiological models typically consider only a finite number ofstrains (or alleles), and thus mutations may cause all strains to bepresent at all times (in mutation–mutation balance) because ofrecurrent mutation from other types. Suchmodels therefore cannot ingeneral meet the population genetic criterion for a neutral null model,and immigration (Lipsitch et al., 2000a,b) or mutation (Bonhoefferand Nowak, 1994) can maintain stable strain coexistence.

Population genetic models usually take a slightly differentperspective, allowing for evolution at the level of genetic sequencesand thus consider many or an infinite number of possible alleles. Inthat case, coexistence of strains is a result of the balance betweenmutation and extinction in a constant process of strain turnover(Fraser et al., 2005). Some models, notably several motivated byinfluenza dynamics, incorporate both population genetic and epide-miological features (Boni et al., 2004; Grenfell et al., 2004; Boni et al.,2006; Koelle et al., 2006; Day and Gandon, 2007).

Discussion

We have shown that any model whose ancestor-tracing ecologicaldynamics can bewrittenwithout naming the strains involvedwill lacka stable coexistence equilibrium for indistinguishable strains. A modelthat satisfies these two criteria is thus an appropriate null model withwhich to attempt to understand coexistence. Thus, if a neutral nullmodel for indistinguishable strains is identified, it may serve as a basis,upon which mechanisms that might promote stable coexistence maybe layered by creating additional states or making decisions that, forexplicit biological reasons, the parameters for two ormore biologicallydifferent strains should depart from those which satisfy the criteria.For example, if biological considerations suggest that distinct strainsare able to colonize distinct niches within a host by attaching todifferent receptors, then one might postulate that Model 13 isappropriate with parameters that violate our criteria for null models:for example, with kN0 (allowing dual infections) but with 0Vcb 1

2(because exposure to one strain is less likely to displace another strainthan to create a dual infection), and/or with 1zqN 1

2 (because by usingdifferent receptors, the two strains can create more total infectious-ness from a coinfected host than if only one strain were present).Alternatively or in addition, strains may experience tradeoffs betweeninfectiousness β and clearance rate u, between different modes oftransmission, or between other properties that could promotecoexistence. The key point is that such hypothetical mechanisms ofcoexistence should be explicit, i.e. layered on top of a model that doesnot predict stable coexistence of indistinguishable strains.

From a different perspective, the analysis here indicates that priorefforts to model the coexistence of strains have had to incorporatemechanisms to permit stable coexistence, often without explicitbiological justification. Lipsitch (Lipsitch, 1997) modeled the

12 M. Lipsitch et al. / Epidemics 1 (2009) 2–13

coexistence of two serotypes of Streptococcus pneumoniae or othercolonizing bacteria using Model 13 but with parameter values q=1,c=0, and kN0. This parameterization violates our criteria, notsurprising in retrospect since the purpose of the earlier model wasto study perturbations to a stable state of coexistence amongserotypes. The model used there is justified in a general sense, bythe fact that in every population studied, one observes the coexistenceof pneumococcal serotypes, often maintained over long periods oftime (Bogaert et al., 2004; Lipsitch and O'Hagan, 2007), and thepurpose of the study was to understand the effects of selectivelyinhibiting one serotype by vaccination. However, the mechanismpermitting stable coexistence was not specified for that model.Different serotypes have many biological differences — includinginvasiveness (Hanage et al., 2005), duration of colonization (Sleemanet al., 2006; Hogberg et al., 2007), accessibility of surface determinants(Abeyta et al., 2003), linkage disequilibrium with antimicrobialresistance (McCormick et al., 2003) and the pilus operon (Basset etal., 2007), and possibly susceptibility to mechanisms of immunitydirected at noncapsular antigens (Malley et al., 2005). Unfortunately,it is not currently clear which of these mechanisms (perhaps it isseveral) accounts for the maintenance of diversity in this species(Lipsitch and O'Hagan, 2007). In the absence of such certainty, ageneric mechanism for promoting stable coexistence in a model – inthis case, the existence of a “protected” class of dually infected persons– may be justifiable as a placeholder but does not represent amechanistic explanation of stable coexistence. Moreover, predictionsabout how the population will change as a result of policy changes,such as vaccination or antimicrobial use, must be regarded as tentativewhen they rely on a model in which stable coexistence of strains priorto intervention is permitted through an unidentified mechanism. Onthe other hand, just as the classic Lotka–Volterra model with noexplicit resource but with phenomenological competition terms forinter- and intraspecific competition provides considerable analyticinsight into competitive dynamics, such “phenomenological” modelsof strain coexistence may be valuable tools until we have a clearerunderstanding of the mechanisms maintaining coexistence.

Such models have been used by a number of investigators besidesthose already cited, for example Zhang and colleagues (Zhang et al.,2007) to study host–pathogen coevolution and Turner and Garnett(Turner and Garnett, 2002) to study multiple strains of gonorrhea.Elbasha and Galvani (Elbasha and Galvani, 2005) used a similar model,though including immune states, to study vaccination against multipleHPV types. In each case, themodel structure builds in stable coexistenceof strains “for free,” without stating an explicit mechanism.

Other authors have built in an explicit mechanism for stablecoexistence of nonidentical strains, such as a tradeoff betweentransmission fitness to uninfected hosts vs. ability to “superinfect” ahost already carrying a the other strain (Levin and Pimentel, 1981;Martcheva et al., 2008). In such cases, the model reduces forindistinguishable strains to Eq. (10), where individuals simply haveone strain or the other and are impervious to superinfection; byallowing one strain to transmit better and the other to superinfectbetter, stable coexistence can be maintained (Levin and Pimentel,1981). Mosquera generalized such models to include both “super-infection” (replacement of one strain by another) and “coinfection”(dual infection with both strains) (Mosquera and Adler, 1998).

A related problem was faced by Austin et al. (Austin et al., 1999),who wished to model the coexistence of drug-sensitive and drug-resistant strains of S. pneumoniae. These authors faced the problemthat a simple model of resistant and sensitive strains with nopossibility for a host to carry both strains (k=0) leads to competitiveexclusion of resistant or sensitive strains, depending on the balancebetween the use of antimicrobial drugs and the fitness deficit ofresistant strains (Lipsitch, 2001a,b).

Austin et al (Austin et al., 1999) created a model that permittedstable coexistence of the two strains by two mechanisms. First, they

allowed for subdivision of the host population into those currently ontreatment and those currently not on treatment. This in principle is aform of “habitat heterogeneity” that in general ecological terms canlead to stable coexistence, though as we show in a companion paper(Colijn et al., 2008) this mechanism alonemakes a minor contributionto stable coexistence of resistant and sensitive strains. In addition, themodel was set up such that although a host could have only one strainat a time, each strain could displace the other by “superinfection”.Austin et al. used parameters such that, in line with that requirement,the sensitive strain was less transmissible than the resistant strain(even in the absence of antimicrobial use), but better at super-infection. To date, we are aware of no evidence that resistant strainsare more transmissible than sensitive ones for any bacterium, andconsiderable evidence to the contrary (Andersson and Levin, 1999;Andersson, 2003). Moreover, the relative rates of superinfection ofdifferent strains are, we believe, entirely unstudied. Nonetheless,Austin et al (Austin et al., 1999) designed a model that could capturethe observed phenomenon of stable coexistence of resistant andsensitive strains, albeit with a mechanism that was not well supportedby empirical data.

A separate problem of ongoing interest is the characterization of thegenetic diversity of pathogens, and what light this can shed onevolutionary and epidemiological processes. The models typicallytreat epidemiological effects as entirely implicit (Fraser et al., 2005;Rambaut et al., 2008), in some cases using discrete generations.fortractability, although the dynamics clearly have overlapping genera-tions (Fraser et al., 2005). In rare cases, the effect of strain immunity isexplicitly included via a coupled epidemiological and populationgenetic model (Boni et al., 2006; Koelle et al., 2006). In this case,where the aim is to identify and characterize selective forces, it isimportant that the nullmodels be neutral; our analysis partly addressesthis, but further work is needed to examine whether stochasticfluctuations in higher dimensional epidemic models can be mappedonto neutral fluctuations in gene frequencies derived for simplepopulation genetic models. This would be needed for example tomodel explicitly the effects of co-infection, recombination andepidemic spread on gene frequencies, all previously treated implicitly(Fraser et al., 2005). It is important to note that we have not exploredthe conditions for a model to generate neutral or nearly neutral driftdynamics amongst strains, which are quite general and parameterdependent, but have rather attempted to elucidate conditions underwhich models may generate selection due to structural features evenwhen all parameters are equal.

In conclusion, we have described two simple criteria that ensurethat null models for strain interactions do not “build in” stablecoexistence through implicit, unstated mechanisms, and have shownthat if the “ancestor-tracing ecological” version of a model satisfies theecological criterion (autonomous ecological dynamics for indistin-guishable strains), then the model itself also satisfies both theecological and population-genetic (no stable coexistence of identicalstrains) criteria. If the goal of a modeling exercise is to understand thebiological mechanisms underlying strain coexistence, then the modelthat includes such mechanisms should reduce – in the case of identicalstrains – to one that satisfies this criterion. When the mechanism ofcoexistence is not the object of study, it may be necessary to build insuch mechanisms in order to create a model that reproduces observedpatterns of coexistence, but ultimately one’s confidence in predictionsofmodels that rely on (butdonot seek to explain) coexistencewould beincreased if one had confidence in the mechanisms of coexistence thatare assumed. An exciting area of ongoing research is to understand theroles of selective (Gupta et al., 1997; Ferguson et al., 2003; Koelle et al.,2006) vs. neutral (Fraser et al., 2005) explanations of persistentpathogen strain diversity for a range of pathogens, and among theselective explanations to distinguish thosebased on acquired immunityto specific and conserved antigens, differential tropisms for tissues andreceptors, and other mechanisms of niche differentiation.

13M. Lipsitch et al. / Epidemics 1 (2009) 2–13

Contributors

ML and CF developed the question addressed by this article, withinput from TC andWPH. ML and CF performed the analysis, with inputfrom CC. ML wrote the first draft of the manuscript, which wasextensively revised by all authors. All authors approved the finalversion of the manuscript.

Acknowledgments

ML was supported by cooperative agreement 5U01GM076497(Models of Infectious Disease Agent Study) from the NationalInstitutes of Health. WPH and CF were supported by Royal SocietyResearch Fellowships. Edward Goldstein is thanked for helpfuldiscussions.

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